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I am trying to prove the following:

Let $I, J$ be ideals in a ring $R$. Prove that the residue of any element of $I\cap J$ in $R/IJ$ is nilpotent.

I am not sure that I have got it right, here is my reasoning:

For any $x\in I\cap J$ we have $x^2\in IJ$. So $x^2+IJ=IJ$. This implies that $x^2=0$ in $R/IJ$.

Is this correct?

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up vote 0 down vote accepted

Yes, it seems that's correct and that's all.

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